Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
[Subject area]
Physics
[Figures and Tables]
10 Figures color
1 table black and white
List of Figures
Fig. 1 The cluster atoms model for Sr58RE6Ti64O192 (X = La, Sm and Er)
Fig. 2 Calculated of lattice constant for Sr0.9RE0.1TiO3 (RE = La, Sm and Er) at
room temperature
Fig. 3 Calculated of lattice constant for Sr0.9X0.1TiO3 (RE = La, Sm and Er) various
temperature
Fig. 4 Calculated of MSD for Sr0.9X0.1TiO3 (RE = La, Sm and Er) various
temperature
Fig. 5 Atoms excursion of Sr, Ti, O, La, Sm and Er at temperature 300 K
Fig. 6 Atoms excursion of Sr, Ti, O, La, Sm and Er at temperature 1100 K
Fig. 7 Pair correlation function (PCF) of (a) OO, TiO and TiTi, (b) SrTi,
LaTi, SmTi and ErTi, (c) SrSr, LaSr, SmSr and ErSr, and (d) SrO,
LaO, SmO and ErO various distance at temperature 300 K
Fig. 8 Pair correlation function (PCF) of (a) OO, TiO and TiTi, (b) SrTi,
LaTi, SmTi and ErTi, (c) SrSr, LaSr, SmSr and ErSr, and (d) SrO,
LaO, SmO and ErO various distance at temperature 1100 K
Fig. 9 Calculated of (a) density and (b) molar volume for Sr0.9RE0.1TiO3 (RE =
La, Sm and Er) various temperature
Fig. 10 Calculated of (a) potential energy, (b) kinetic energy and (c) entropy for
Sr0.9RE0.1TiO3 (RE = La, Sm and Er) various temperature
List of table
TABLE 1: Interatomic potential function parameter for Sr58RE6Ti64O192 (RE = La,
Sm and Er)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Meena Rittiruam1, 2, Kunchit Singsoog1, 2 ,
Hassakorn Wattanasarn1, 2, Athorn Vora-ud1, 2,
Suwipong Hemathulin3 and Tosawat Seetawan 1, 2, *
1
Simulation Research Laboratory, Thermoelectrics Research Center,
Research and Development Institution, Sakon Nakhon Rajabhat University,
680 Nittayo Road, Mueang District, Sakon Nakhon, 47000, Thailand
2
Program of Physics, Faculty of Science and Technology, Sakon Nakhon
Rajabhat University, 680 Nittayo Road, Mueang District, Sakon Nakhon,
47000, Thailand
3
Program of Mechanical and Industrial, Faculty of Industrial Techonology,
Sakon Nakhon Rajabhat University, 680 Nittayo Road, Mueang District,
Sakon Nakhon, 47000, Thailand
E-mail addresses:
meena_physics-snru@yahoo.co.th (Meena Rittiruam)
kunchitsingsoog@yahoo.com (Kunchit Singsoog)
w_hussakorn@hotmail.com (Hassakorn Wattanasarn)
a_thorn2008@hotmail.com (Athorn Vora-ud)
acumajo@gmail.com (Suwipong Hemathulin)
t_seetwan@snru.ac.th (Tosawat Seetawan)
*Corresponding Author
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
ABSTRACT
Strontium titanate substitute of lanthanide metals Sr0.9RE0.1TiO3 (RE = La,
Sm and Er) were simulated by molecular dynamics (MD) method in
temperature range 300 K – 1100 K to evaluate the quantum properties viz.,
lattice constant, density (), molar volume (Vm), mean square displacement
(MSD), pair correlation function (PCF), potential energy (U), kinetic energy
(Ek) and entropy (S). In MD calculation, we define initial position of 320
atoms (O = 192, Ti = 64, Sr = 58 and X = 6) base on Perovskite structure in
cluster atoms site of 4×4×4. The interatomic interaction used the
Morsetype potential functions added to the Busing–Ida potential, which
composed of Coulomb interactions term, short range repulsion term, van der
Waals attraction term and covalent term. We are calculated force by Newton
equation of motion, and calculate atom positions and velocities by Verlet’s
algorithm. And then evaluate the energy by Ewald’s summation to evaluate
quantum properties. It was found that, the lattice constant was decrease form
3.905 Å to 3.8959 Å, 3.8996 Å and 3.8990 Å when substituted of La, Sm
and Er, respectively. The PCF of SrO, SrTi and SrSr show that decrease
with substituted of La, Sm and Er. Furthermore, the PCF was decreased
with increasing temperature. The lattice constant, molar volume, mean
square displacement, energy and entropy also increased with increasing
temperature, while density decreased and correspond with literature data.
The quantum properties of these materials are interesting for feature study
thermal properties.
Keywords: lattice constant, mean square displacement, pair correlation
function, Busing−Ida potential, Morse−type potential, MXDORTO, thermo
dynamics, thermoelectric materials
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
INTRODUCTION
Strontium titanate (SrTiO3) base on cubic perovskite structure with space
group number is 221, space group symbol is Pm 3 m , lattice constant
𝑎 = 3.9050 Å at room temperature (Richet, 1996). Since, SrTiO3 is a
n-type oxide thermoelectric (TE) material with a large thermopower
> 0.1 mV K1, thermal conductivity of 8 W m1 K1 and ZT of 0.08 at room
temperature (Rowe, 2006). To development of TE performance could be
several strategies substitutional solid solution formation. The rare earth (RE)
was substituted by formula Sr0.9RE0.1TiO3 (RE = Nd, Sm, Gd, Dy, Y, Er,
Yb, La) shown that Er have a low thermal conductivity, and Sm high ZT at
1100 K, respectively (Liu, 2013). In addition, the lattice constant was
reducing after substitute RE and thus reduces the lattice thermal
conductivity (Shang, 2010 and Liu, 2013). Recently, we successes useful
molecular dynamics method simulated thermal properties of perovskite
structure i.e., SrTiO3 (Seetawan, 2010), Ca0.8M0.2MnO3 (M = Cu; Ag; and
Bi) (Seetawan, 2014) and Ca1-XEuXMnO3 (X = 0, 0.05, 0.10, 0.15)
(Rittiruam, 2014), which also main focus on thermal conductivity. However,
we do not analysis on the thermal properties of crystal such as pair
correlation function, mean square displacement, density, molar volume,
energy, until the entropy. In this work, we interest in the TE properties of
SrTiO3 substitute lanthanides metal. Firstly, we would like study on thermal
properties of these material by using computer to simulation before
experiment. Which, we choose La, Sm and Er for substitute by formula
Sr0.9RE0.1TiO3 to simulation by molecular dynamics method for study
thermal properties.
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
COMPUTATIONAL DETAILS
The cluster atoms model of Sr0.9X0.1TiO3 were designed by using 320 atoms
viz., O = 192, Ti = 64, Sr = 58 and X = 6 atom, respectively, based on
perovskite stucture, as show in Fig. 1.
In MD calculation, Firstly solving the force for all atomic in the cluster
atom by using Newton equation of motion, following eq. (1);
Fi  mi
 2 ri
t 2
=
U (ri ,..., rN )
ri
; i  1,..., N
(1)
where mi , ri , t and U are mass of atom i th , position of atom i th , time and
potential energy function, respectively.
Next, we calculate the positions and velocities in these cluster atoms by
using Verlet’s algorithm (Verlet, 1967) base on MXDORTO program
(Kawamura, 1994). Time step of 2.0×1015 s was used in calculating
velocity. The scaling method was used to control of temperature and
pressure. Which, we calculated these cluster atoms in temperature rang
300 K1100 K, pressure in 1 MPa. Then, the energy was evaluated by
Ewald’s summation (Wigner, 1932). The cationanion interactions used
include of Coulomb interactions, short range repulsion, van der Waals
attraction and covalent term. The Morsetype potential function (Morse,
1929) was added in BusingIda potential function ( Ida, 1 9 7 6 ) for
interatomic interaction, as given by eq. (2);
U ij (rij ) 
zi z j e2

rij
ci c j
rij6
 ai  a j  rij 
 f 0 (bi  b j )exp 

 bi  b j 
 Dij exp  2 ij (rij  rij )   2exp   ij (rij  rij )  (2)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
where f0 , zi , z j , rij and rij* , are repulsion betaween atom in vacuum equal
4.186, the effective partial electronic charges on the i th ions, the effective
partial electronic charges on the j th ions, inter–atomic distance, bond length
of the cation–anion pair in vacuum. a, b and c are the characteristic
parameters depending on the ion species. Dij and  ij , describes the depth
and shape of this potential, respectively.
The interatomic potential function parameters for these clusters atomic were
carries out, as show in table 1.
RESULTS AND DISCUSSIONS
The lattice constant of Sr0.9La0.1TiO3, Sr0.9Sm0.1TiO3 and Sr0.9Er0.1TiO3 were
carried out by MD calculation with value 3.89590.0023 Å, 3.89960.0022 Å
and 3.89900.0021 Å at room temperature, respectively. Comparison of
calculated with experiment of the lattice constant shown structure decreased
upon substituted La, Sm and Er, which in general of SrTiO3 have lattice
constant 3.905 Å at room temperature (Richet, 1996). Another, the lattice
constant of these calculated were compared with the calculated result from
MXDORTO of SrTiO3 (Seetawan, 2010). It was found that, with increasing
temperature the lattice constant of all calculated also increased. The
calculated
lattice
constant
of
Sr0.9La0.1TiO3,
Sr0.9Sm0.1TiO3
and
Sr0.9Er0.1TiO3 shows in Fig. 2 and 3.
Mean square displacement (MSD) could be calculated from
expectation of displacement at initial time to total time, by following
equations;
MSD  [r (t )  r (0)]2  r (t )2  r (0)  2r (0) r (t )
(3)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
where, r (t ) and r (0) displacement at time t and displacement at initial time,
respectively.
Figure 4 shows that the atoms of oxygen can large movement when
compared with other atoms at room temperature, and then these atoms are
tend constant vibration when increasing temperature. On other way, Ti
atoms are short movement when compared of other atoms in same
temperature, and then it’s tend constant vibration same of oxygen atom.
Because of O and Ti were many interaction i.e., TiO, SrO, (La, Sm,
Er)O OO, SrTi and (La, Sm, Er)Ti. Moreover, Ti it’s live in centred of
the unit cell of SrTiO3, could saw previous in Fig. 1. In addition, the Sr
atoms are tending movement quite independently when increasing
temperature. After La, Sm and Er were substituted Sr shown increase MSD
more than Sr, it’s indicated that of the substitution of lanthanides atom
influence to movement and vibration. All of the about movement and
vibration were easily descripted by excursion of atoms in unit cell in as
show in Fig. 5 and 6. The pair correlation function (PCF) has been
descripted by equation (4) (Lowden, 1973);
g (r )   2 
N
  (ri ) (rj  r )
(4)
i  j 1
where  and  (r ) are density and Dirac delta function. Which the PCF was
carried out by MXDORTO. We found that, bonding of TiO, OO and
SrTi were indicated explicitly crystallinity, because these bond shown
distance of bond by total peak, denote the orderly atomic arrangement. In
other peak shown the atoms are neighbours, especially bonding of (Sr, La,
Sm, Er)O. By the way, after substitute of La, Sm and Er its shown
decrease of total peak of bonding LaO, SmO and ErO, it’s indicated that
these substituting affect to decrease crystallinity. In addition, at temperature
1100 K the bonding of ErTi is neighbours by in about rang 1.5 Å2.5Å ,
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
and then the bonding of ErO is clearly increase distance from about 1.5 Å
at 300 K to about 2.5 Å at 1100 K. Moreover, at above temperature also
affect to decrease crystallinity. The PCF of Sr0.9X0.1TiO3 (X = La, Sm and Er)
at temperature 300 K and 1100 K shows in Fig. 7 and 8, respectively. The
calculated of density of substituted La, Sm and Er had a value
5.288 g cm3, 5.303 g cm3 and 5.40 g cm3 at atm, respectively.
Figure 8(a) shows calculated density compared with theoretical density,
which these calculate are agreeing with data of Liu (Liu, 2013). We found
that, the element of substitute contribution to change the density, it’s
dependent on atomic radius and mass. Hence, in case of substitute Er is
density more than La and Sm. Moreover, the density and molar volume
were compared with temperature, shown density was decreased but molar
volume increased with increasing temperature. Hence, the structure had
expanded when with increasing temperature. Figure 10 shows the calculated
of potential energy, kinetic energy and applied to calculate entropy. The
potential was calculated by sum of Coulomb energy and short rang energy,
given by;
Utotal  UCoulomb  U short  UCoulomb  UvdW  U srr  U Morse
(5)
where U Coulomb , U vdW , U srr and U Morse are Coulomb potential, van der
Waals attraction, short range repulsion and Morse potential energy,
respectively. The kinetic energy and entropy can evaluate by equations;
N
Ek 
 mi vi2
i
3 Nk B
H  Etotal  PV  Ek  Utotal  PV
and;
PV 
2
2 N
Ek  Nk BT   mi vi2
3
3 i
(6)
(7)
(8)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
N
Hence; H  U Coulomb  U vdW  U srr  U Morse 
 mi vi2
i
3Nk B

2 N
 mi vi2
3 i
(9)
where H , m , v , N and k B are entropy, mass, velocity, total of atom and
Boltzmann constant, respectively. It was found, the potential energy of
substitute of La, Sm and Er hardly non difference, the kinetic energy also
same value in various temperatures. The energy contribution to entropy of
system, these work found that entropy little lower potential energy
CONCLUSION
The MD was used calculate Sr0.9RE0.1TiO3 (RE = La, Sm and Er) to
evaluate thermal properties. The thermal properties composed of lattice
constant, MSD, PCF, , Vm, U, Ek and S. The lattice constant at room
temperature of substitute La, Sm and Er are 3.89590.0023 Å,
3.89960.0022 Å and 3.89900.0021 Å, respectively, were carries out by
MD calculation. Which agree with the data experiment of El-Mallah (ElMallah, 2007) and Liu (Liu, 2013). The calculated of MSD, PCF,  and Vm
had descripted the crystal structure of SrTiO3 decrease after substituted of
La, Sm and Er, its according with data of lattice constant. However, the
energy and entropy of system do not change. We found the lattice constant,
MSD, Vm, U, Ek and S also increased while  and PCF decreased at
increasing temperature. The quantum properties of these materials are
interesting for feature study thermal properties.
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
REFERENCES
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Mechanisms Studies in Strontium Lanthanum Titanate Perovskite.
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computer. Tokyo: Shokabo.
Liu, J., Wang, C. L., Li, Y., Su,W. B., Zhu, Y. H., Li, J. C., & Mei, L. M.
(2013). Influence of rare earth doping on thermoelectric properties
of SrTiO3 ceramics. Journal of Applied Physics, 114, 223714.
Lowden, L., Chandler, D. (1973). Solution of a new integral equation for
pair correlation function in molecular liquids. The Journal of
Chemical Physics, 59(12), 6587-6595.
Morse, P. M. (1929). Diatomic Molecules According to the Wave
Mechanics. II. Vibrational Levels. Physics Review, 34, 57-65.
Richet, P., & Ligny, P. D. de,. (1996). High-temperature heat capacity and
thermal expansion of SrTiO3 and SrZrO3 perovskites. Physical
Review B, 53, 3013-3022.
Rittiruam, M., Wattanasarn, H., & Seetawan, T. (2014). Thermophysical
Properties of Ca1-XEuXMnO3 (X = 0, 0.05, 0.10, 0.15) Simulated by
Classical Molecular Dynamics Method. Chiang Mai University
Journal of Natural Sciences, 13, 585-593.
Rowe, P. D. D. M., (2006). Thermoelectrics Handbook Marco to nano.,
(p. 35-6). New York: Taylor & Francis
Seetawan,T., Wong ud-dee, G., Thanachayanont, C., & Amornkitbumrung,
V. (2010). Molecular Dynamics Simulation of Strontium Titanate.
Chinese Physics Letters, 27(2), 026501.
Seetawan, T. (2014). Theoretical Analysis of the Substitutable Metal on the
Thermoelectric Performance of CaMnO3. Integrated Ferroelectrics,
155, 9-14.
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Shang P., Zhang, B., Li, J., & Ma, N. (2010). Effect of sintering temperature
on thermoelectric properties of La-doped SrTiO3 ceramics prepared
by solegel process and spark plasma sintering. Solid State Sciences,
12, 1341-1346.
Verlet, L. (1967). Experiments on Classical Fluids. I. Thermodynamical
Properties of Lennard Jones Molecules. Physics Review, 98-103.
Wigner E., (1932). On the Quantum Correction for Thermodynamic
Equilibrium. Physics Review, 40, 749-759.
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Fig. 1 The cluster atoms model for Sr58RE6Ti64O192 (X = La, Sm and Er)
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
3.904
Lattice constant (Å)
3.902
H.M.El-Mallah (2007)
J. Liu et al. (2013)
3.900
3.898
3.896
3.894
T = 300 K, P = 1 MPa
3.892
La
Sm
Er
Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Fig. 2 Calculated of lattice constant for Sr0.9RE0.1TiO3 (RE = La, Sm and Er) at
room temperature
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
3.94
Lattice constant (Å)
3.93
SrTiO3
3.92
3.91
3.90
3.89
La
T. Seetawan (2010)
3.88
500
1000
Sm
500
1000
Er
500
1000
Temperature (K)
Fig. 3 Calculated of lattice constant for Sr0.9X0.1TiO3 (RE = La, Sm and Er) various
temperature
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
0.5
2
MSD (Å )
0.4
O
0.3
0.2
Sr
0.1
0.0
La
Sm
Er
Ti
500
1000
500
1000
Temperature (K)
Fig. 4 Calculated of MSD for Sr0.9X0.1TiO3 (RE = La, Sm and Er) various
temperature
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
La
Sr
Sm
Sr
Ti
O
Ti
O
300 K
Er
Sr
Ti
O
300 K
300 K
Fig. 5 Atoms excursion of Sr, Ti, O, La, Sm and Er at temperature 300 K
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
Sr
La
Sm
Sr
Ti
Ti
O
O
1100 K
Er
Sr
Ti
O
1100 K
1100 K
Fig. 6 Atoms excursion of Sr, Ti, O, La, Sm and Er at temperature 1100 K
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
2500
800
(a)
2000
600
(b)
500
1500
400
O-O
Ti-O
Ti-Ti
1000
PCF at T = 300 K
Sr-Ti
La-Ti
Sm-Ti
Er-Ti
700
300
200
500
100
0
1.5
2.0
2.5
3.0
3.5
4.0
350
300
250
0
2.5
250
Sr-Sr
La-Sr
Sm-Sr
Er-Sr
3.0
3.5
Sr-O
La-O
Sm-O
Er-O
200
4.0
(d)
150
200
150
100
100
50
0
3.0
50
(c)
3.2
3.4
3.6
3.8
4.0
0
1
2
3
4
Distance (Å)
Fig. 7 Pair correlation function (PCF) of (a) OO, TiO and TiTi, (b) SrTi,
LaTi, SmTi and ErTi, (c) SrSr, LaSr, SmSr and ErSr, and (d) SrO,
LaO, SmO and ErO various distance at temperature 300 K
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
2500
800
(a)
2000
600
(b)
500
1500
PCF at T = 1100 K
Sr-Ti
La-Ti
Sm-Ti
Er-Ti
700
400
O-O
Ti-O
Ti-Ti
1000
300
200
500
100
0
1.5
2.0
2.5
3.0
3.5
4.0
300
250
0
1
250
350
Sr-Sr
La-Sr
Sm-Sr
Er-Sr
(c)
200
200
150
150
100
2
3
Sr-O
La-O
Sm-O
Er-O
4
(d)
100
50
50
0
2.5
3.0
3.5
4.0
0
1
2
3
4
Distance (Å)
Fig. 8 Pair correlation function (PCF) of (a) OO, TiO and TiTi, (b) SrTi,
LaTi, SmTi and ErTi, (c) SrSr, LaSr, SmSr and ErSr, and (d) SrO,
LaO, SmO and ErO various distance at temperature 1100 K
(b)
36.4
3
–3
Density (g cm )
5.35
36.6
–1
(a)
5.40
Molar volume (cm mol )
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
5.30
5.25
5.20
5.15
La
Sm
Er
Liu et al. (2013)
Liu et al. (2013)
500
36.2
36.0
35.8
La
Sm
Er
35.6
1000
500
1000
Temperature (K)
Fig. 9 Calculated of (a) density and (b) molar volume for Sr0.9RE0.1TiO3 (RE =
La, Sm and Er) various temperature
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
–1
70
La
Sm
Er
La
Sm
Er
-6040
-6050
-6060
(a)
-6070
500
1000
–1
-5960
60
–1
-6030
Entropy (J mol K )
–1
-6020
-5940
La
Sm
Er
Kinetic energy (J mol K )
–1
–1
Potential energy (J mol K )
-6010
50
40
-5980
-6000
-6020
(c)
(b)
30
500
1000
-6040
500
1000
Temperature (K)
Fig. 10 Calculated of (a) potential energy, (b) kinetic energy and (c) entropy for
Sr0.9RE0.1TiO3 (RE = La, Sm and Er) various temperature
Quantum properties of Sr0.9RE0.1TiO3 (RE = La, Sm and Er)
TABLE 1: Interatomic potential function parameter for Sr58RE6Ti64O192 (RE = La,
Sm and Er)
Atom
O
Ti
Sr
La
O
Ti
Sr
Sm
O
Ti
Sr
Er
TiO
SrO
LaO
SmO
ErO
z
a (Å)
b (Å)
For Sr58La6Ti64O192
1.9232
0.16
1.2
1.2
1.055
0.18
1.2
1.198
0.16
1.2
0.6
0.16
For Sr58Sm6Ti64O192
1.926
0.16
1.2
1.2
1.055
0.18
1.2
1.198
0.16
1.2
0.6
0.16
For Sr58Er6Ti64O192
1.9256
0.16
1.2
1.2
1.055
0.18
1.2
1.198
0.16
1.2
0.6
0.16
Atom pair
19

D (10 J)
β (Å1)
4.3
3.82
2.41
1.18
2.60
1.18
2.60
1.18
2.60
1.18
c (kJ1/2 Å3
mol1/2)
20
25
10
0
20
25
10
0
20
25
10
0
r* (Å)
2.1923
2.7615
2.7615
2.7615
2.7615